1. Field of the Invention
This invention relates to an adaptive controller with a parameter adjustment law expressed in a recurrence formula(e), more particularly to an adaptive controller with a parameter adjustment law expressed in a recurrence formula(e) proposed by I. D. Landau et al, in which the range of change of intermediate values, i.e., the range within which the values can possibly change, of the parameter adjustment law is limited to such an extent that the adaptive controller can be realized on a low-level microcomputer with fewer bit (word) length such as 16-bits.
2. Description of the Prior Art
Among adjustment laws used in adaptive controls, there is one known law proposed by I. D. Landau et al to be used in model reference adaptive control systems (MRACS) or self-tuning regulators (STR). In the proposed technique, an adaptive system is converted into a feedback system equivalent thereto, comprised of a linear block and a non-linear block. The adjustment law is determined such that the input and output of the non-linear block satisfy Popov's integral inequalities, while the transfer function of the linear block becomes perfect positive real, thereby ensuring system stability. The technique was described in "COMPUTROL", No. 27, pp.28-41 (Tokyo: CORONA Publishing Co., Ltd.), or Automatic Control Handbook, pp.703-707 (Tokyo: Ohm Publishing Co., Ltd., 1983).
The parameter adjustment law proposed by I. D. Landau et al will further be explained with reference to a simple STR illustrated in FIG. 1. Here, polynomials of the numerator and denominator B/A (the plant--the controlled object of the system) are defined as (1)(2) of Equation 1. The parameter or parameter vector of the adaptive controller .theta.(k) and the input zeta (k) to the parameter adjuster are defined as in (3) and (4) of Equation 1. ##EQU1##
.theta.(k) is also expressed in Equation 2. .GAMMA.(k) and e*(k) in Equation 2 represent gain matrix and identification error signal, and are expressed in recurrence formulae as shown in Equations 3 and 4. EQU .theta.(k)=.theta.(k-1)+.GAMMA.(k-1).xi.(k-d) e*(k) Eq. 2 ##EQU2##
Depending on how .lambda..sub.1 (k) and .lambda..sub.2 (k) are chosen in Equation 3, one algorithm is selected. For example, if .lambda..sub.1 (k)=1 and .lambda..sub.2 (k)=0, it is a constant gain algorithm. If .lambda..sub.1 (k)=1 and .lambda..sub.2 (k)=.lambda.(0&lt;.lambda.&lt;2), it is a gradually-decreasing gain algorithm (least squares method when .lambda.=1). If .lambda..sub.1 (k)=.lambda..sub.1 (0&lt;.lambda..sub.1 &lt;1) and .lambda..sub.2 (k)=.lambda..sub.2 (0&lt;.lambda..sub.2 &lt;.lambda.), it is a variable gain algorithm (weighted least squares method when .lambda..sub.2 =1). When rewriting .lambda..sub.1 (k)/.lambda..sub.2 (k) as .sigma. and .lambda..sub.3 is expressed as in Equation 5, if .lambda..sub.1 (k)=.lambda..sub.3, it is a constant trace algorithm. ##EQU3##
Aside from the above, when such a calculation is conducted on a digital computer, since the digital computer has a particular bit (word) length such as 4, 8, 16 or 32 bits intrinsically determined by the configuration of its registers, data buses or the like, variables in the calculation have to be restricted, in some cases, within a predetermined range. For example, when a variable's least significant bit (LSB) is assigned with a small value so as to enhance calculation accuracy, a possible maximum value of the variable is thereby limited to a certain extent, and hence the range of change of the variable is automatically restricted. If the parameter adjuster is intended to be realized on a digital computer, not all variables of the parameter adjuster are free from the problem. Note that the "variable" or "intermediate variable" referred to in the specification means all variables calculated finally (or even temporarily) in the adaptive controller such as .GAMMA.(k), .GAMMA.(k-1).xi.(k-d),.GAMMA.(k-1).xi.(k-d).GAMMA..sup.T (k-1), .theta..sup.T (k-1).xi.(k-d), etc.
This will be explained with reference to a time-varying plant illustrated in FIG. 2. The configuration in FIG. 2 is a modification of the STR illustrated in FIG. 1; specifically, in the STR illustrated in FIG. 1, some values are defined more concretely as: EQU delay d=2, EQU A(z.sup.-1)=1+a.sub.1 z.sup.-1,
and EQU B(z.sup.-1)=b.sub.0 +b.sub.1 z.sup.-1,
wherein the values a.sub.1, b.sub.0, b.sub.1 are supposed to change with respect to time. The model of the configuration in FIG. 2 is assumed as expressed in Equation 6. ##EQU4##
Assume that each variable's LSB is assigned with a sufficiently small value and a simulation is conducted with sufficient bit length. If such an input r(k) as is expressed in FIG. 3 is given to the controller, the output y(k) from the plant will be as shown in FIG. 4. FIG. 5 shows the control error (=r(k)-y(k)), and demonstrates that the STR follows the plant's change and functions properly as a regulator. When selecting one element .GAMMA..sub.11 (k) (first row, first column) of the gain matrix as the representative of the intermediate variables of the parameter adjuster, the value will be as shown in FIG. 6. It will be understood from the figure that the change of the value is relatively large, and ranges from 4.times.10.sup.7 through 8.times.10.sup.8. In integer calculation, it is difficult for a microcomputer with a small string of bits such as 16-bits to handle such a large value.
Further, assume that the bit length is sufficiently prepared for the intermediate values used in the calculation of the parameter adjuster, but that the LSB is assigned with a relatively large value. The control error r(k)-y(k) at that instance is then observed. As illustrated in FIG. 7, the control error becomes greater, degrading control performance. Thus, when the adaptive controller is intended to be realized on a digital computer with high accuracy, it becomes necessary to assign a small value even for the intermediate values' LSBs and if doing so, it becomes necessary to use a digital computer having bit length sufficiently for preventing overflow in the computer. As a result, it is quite difficult to introduce an adaptive controller into a control system having a short sampling period using a low-level, fewer-bit-length, and hence less expensive microcomputer. In other words, if a sampling period in the control system is short, it becomes necessary to use an expensive, high-level microcomputer with larger bit length.
In other words, the method proposed by I. D. Landau et al was a theoretical one, and realizing an adaptive controller on a computer in real time basis and with a short sampling period has not been examined.